**Abstract**:

**Abstract**
Ernst Mach observed that light or dark bands could be seen at abrupt changes of luminance gradient in the absence of peaks or troughs in luminance. Many models of feature detection share the idea that bars, lines, and Mach bands are found at peaks and troughs in the output of even-symmetric spatial filters. Our experiments assessed the appearance of Mach bands (position and width) and the probability of seeing them on a novel set of generalized Gaussian edges. Mach band probability was mainly determined by the shape of the luminance profile and increased with the sharpness of its corners, controlled by a single parameter (*n*). Doubling or halving the size of the images had no significant effect. Variations in contrast (20%–80%) and duration (50–300 ms) had relatively minor effects. These results rule out the idea that Mach bands depend simply on the amplitude of the second derivative, but a multiscale model, based on Gaussian-smoothed first- and second-derivative filtering, can account accurately for the probability and perceived spatial layout of the bands. A key idea is that Mach band visibility depends on the ratio of second- to first-derivative responses at peaks in the second-derivative scale-space map. This ratio is approximately scale-invariant and increases with the sharpness of the corners of the luminance ramp, as observed. The edges of Mach bands pose a surprisingly difficult challenge for models of edge detection, but a nonlinear third-derivative operation is shown to predict the locations of Mach band edges strikingly well. Mach bands thus shed new light on the role of multiscale filtering systems in feature coding.

*Mach bands*, though Weale (1979) argued that the bands were almost certainly known to artists of the 15

^{th}century Italian Renaissance. The bands are typically seen at abrupt changes of luminance gradient, such as the junction of a luminance ramp and a plateau—a luminance profile often known as the

*Mach ramp*. Importantly for theories of visual feature detection (Morgan, 2011), the existence of Mach bands shows that not all perceived bars or lines arise from peaks or troughs in the luminance profile. Given its long history, it is not surprising that quite a number of models and theories of the Mach phenomenon have been proposed (Pessoa, 1996). Though they differ in scope and detail, one central idea—spatial filtering by even-symmetric, center-surround receptive fields—is common to all models, including MIRAGE (Morgan & Watt, 1997; Watt & Morgan, 1985) and the local energy model (Ross, Morrone & Burr, 1989). A long-standing difficulty has been in constructing a broader theory that embeds this filtering into a satisfactory, more general theory of spatial vision. Our aims in this paper are (a) to describe a multiscale model for visual (1-D) feature detection based on earlier models of edge coding (Georgeson, May, Freeman, & Hesse, 2007), extended to deal with both even- and odd-symmetric features (bars and edges) in an integrated fashion and (b) to apply this model to new experimental data on the perceived structure of Mach bands and the probability of reporting them under different conditions.

*scale*and

*derivative*are used extensively in this paper, and we use them in several contexts that should not be confused.

*σ*. Doubling or halving

*σ*alters their size or scale accordingly. But the spatial filters applied to these images also have a range of sizes or filter scales

*s*; doubling or halving

*s*shifts the filter by one octave towards lower or higher spatial frequencies. Importantly, when an image is spatially filtered at multiple scales (a multiscale filter system), then the response to a single stimulus of scale

*σ*is distributed over many filter scales (

*s*) and over space (

*x*). The 2-D distribution of responses over (

*x, s*) is a

*scale-space map*. (For a wide-ranging account of the scale-space approach to biological and computer vision, see ter Haar Romeny, 2003, and for more mathematical detail, see Lindeberg, 1994.)

*Blur*is a simple form of low-pass spatial filtering, in which the image is smoothed by convolution with a (usually) unimodal blur kernel or point-spread function. In Gaussian blur, the kernel is a Gaussian function

*G*(

*x, b*) of unit area (in 1-D) or unit volume (in 2-D). The degree of blurring is controlled by the scale parameter or blur

*b*.

*gradient*profile of a 1-D image whose luminance profile is

*L*(

*x*) is its first derivative

*L′*(

*x*) or

*dL/dx*defined in the usual way. It measures the slope of the function at each point

*x*—the change

*δL*in

*L*over an infinitesimal distance

*δx*. The second derivative is the gradient of the gradient,

*d*

^{2}

*L/dx*

^{2}equal to

*dL′/dx*. It is closely related to local curvature in

*L*(

*x*).

*s*, the combined operation is the

*Gaussian derivative*and the filter kernel is

*dG*(

*x, s*)

*/dx*, sketched at the top of Figure 1D. In the absence of nonlinearity, several blurring and derivative operations can be chained (in any order) to create higher order Gaussian derivatives. Gaussian-derivative filters are spatial frequency tuned filters whose bandwidth decreases as the derivative order (first, second, etc.) increases. Peak spatial frequency is inversely proportional to filter scale

*s*. Receptive field symmetry matches the derivative order: odd orders (1, 3, 5…) have odd symmetry; even orders (2, 4, 6…) have even symmetry. In this paper we consider mainly the first (odd) and second (even) Gaussian derivatives. If we blur an image with Gaussian blur

*b*and compute its Gaussian derivative at scale

*s*, the blurring and smoothing simply combine quadratically,

*s*′ = $ b 2 + s 2 $, and the outcome is exactly equivalent to applying a Gaussian derivative with a larger scale

*s*′. Because of this quadratic relation, small input blurs

*b*have little influence at large filter scales, that is

*s*′ ≈

*s*, if

*b*≪

*s*.

*s*. We try to distinguish clearly between stimulus scale (

*σ*) and filter scale (

*s*). In particular, the scale

*s*of the most active filter can vary with the shape of the waveform as well as the stimulus scale parameter

*σ*.

*scale normalization*(the setting of filter gains) to achieve automatic, image-driven

*scale selection*. This peak-finding scheme can identify the location, scale, and identity of image features—a crucial step in early visual coding. Georgeson et al. (2007) showed how these principles could be used to model the process of localizing blurred edges and encoding edge blur in human vision. They evaluated two models called N1

*+*and N3

*+*that used multiscale first- and third-derivatives, respectively. (Note

*:*N denotes use of Lindeberg's scale-normalization; one or three indicates the order of derivatives used; + indicates the use of half-wave rectification on the output of the filters.) In the present paper we describe how this approach via Gaussian derivatives in scale-space can be extended to encode both bars and edges using even- and odd-symmetric filters, respectively (Figure 1D; described in detail later). We suggest that these filters do not act as independent channels, nor are their responses combined (as in the local energy model; Morrone & Burr, 1988; Ross et al., 1989), nor do they act competitively. Rather, it is a comparison of the even and odd filter responses that enables a decision about feature presence or absence. Such a comparison also formed part of the local energy model where it was used to evaluate local phase and classify energy peaks as either bars or edges. Here we make no use of the energy measure because it is (by definition) the smooth spatial envelope of the even and odd responses, and that turns out to exclude too many features that are actually perceived (Hesse & Georgeson, 2005).

*generalized Gaussian*profile (described below). In Experiments 1 and 3 we used the yes-no method to evaluate the probability of reporting Mach bands, and in Experiment 2 we used the feature-marking method (Georgeson & Freeman, 1997) to assess the geometry of Mach bands by marking both the center positions of the bands and the edges of the bands.

*L*(

*x*) of the vertical 1-D images could vary from a Mach ramp through to a Gaussian edge and beyond under the control of a single shape parameter

*n*. Their first derivative (gradient) profile was defined as a generalized Gaussian function (Figure 2A) where

*n*= 1, 1.5, 2, 2.5, 3, 4, or 5, and

*A*is a constant that controls the gradient magnitude and contrast of the image. When

_{n}*n*= 2 the first derivative was a Gaussian. Each waveform was sampled at 1 min arc intervals and integrated numerically using the cumtrapz function in Matlab and scaled to a common amplitude to form the luminance profile of a blurred vertical dark-to-light edge (Figure 2B). A copy of each waveform was left-right reversed to form a light-to-dark edge. Edges of different scales were obtained by setting

*σ*= 3, 6, or 12 min arc. Increases (or decreases) of

*n*sharpen (or blur) the upper and lower corners of the waveform as shown in Figure 2B. Thus the family of luminance waveforms ranged from a very smooth profile (

*n*= 1; Figure 2C) through the Gaussian integral (

*n*= 2), to a slightly blurred Mach ramp (

*n*= 5; Figure 2D).

*n*= 1, 2, 3, and 5) at the three scales

*σ*, while Figure 3B shows the corresponding second derivative profiles. Figure 3C reveals that the second derivative peak amplitude is almost directly proportional to the exponent

*n*but falls as the inverse-square of scale

*σ*; hence the second derivatives at Scale 12 are 16 times lower than at Scale 3.

*A*) at

_{n}*x*= ±

*σ*. For the edge scales

*σ*= 3, 6, 12 min arc, the corresponding cosine half-periods were 10.25, 20.5, and 41 min arc.

^{2}). At each of the three scales

*σ*, there were 16 test images (eight waveforms, two polarities). Two examples are shown in Figure 2C and 2D.

*n*= 5 waveform) flanked by two plateaux—was shown prior to data collection, as training. All 48 images (16 images at three scales) were shown five times in randomized order within one experimental session, which took about 10 minutes. Six observers completed seven sessions each, but the first session was discarded as practice. Only two observers (the authors) were aware of the stimulus design. All observers gave informed consent.

*σ*= 6 min arc) but with (a) three contrasts (0.2, 0.4, 0.8) at a presentation duration of 300 ms and (b) three presentation durations (50, 100, 300 ms) at a contrast of 0.4. There are only five different conditions here, and these five were presented in separate, randomly interleaved blocks of trials. The exponent

*n*varied randomly from trial to trial (

*n*= 1 to 5) as before. There were five observers, including four who had taken part in Experiment 1. When pooled over the two polarities, there were 60 trials per condition per observer, as in Experiment 1.

*n*. Observers' responses were similar for the two contrast polarities, and so group mean data, averaged across polarity, are shown in Figure 4. The probability of Mach band perception increased smoothly and monotonically as the generalized Gaussian exponent increased.

*F*(1, 5) = 4.9,

*p*= 0.078, while the effect of exponent (

*n*) was highly significant,

*F*(6, 30) = 107.6,

*p*< 0.0001. The main effect of spatial scale

*σ*was not significant,

*F*(2, 10) = 2.47,

*p*= 0.13, but the interaction between scale and exponent that is apparent in Figure 4 was statistically significant,

*F*(12, 60) = 2.09,

*p*= 0.03. Selective analyses, with either Scale 3 or Scale 12 excluded, showed that this Scale × Exponent interaction arose between Scales 3 and 6,

*F*(6, 30) = 4.15,

*p*= 0.004, while there was no interaction between Scales 6 and 12,

*F*(6, 30) = 0.79, n.s. No other effects or interactions were significant. Thus to a large extent the occurrence of Mach bands was scale-invariant, but there was a tendency at the smaller exponents for the smallest-scale stimuli to give more Mach bands than the larger scales did, while Scale 6 gave the most Mach bands at the high exponents. The strong influence of

*n*and the weak, nonsignificant influence of stimulus scale

*σ*imply that the shape of the luminance waveform, rather than its size, is the key factor in Mach bands.

*n*= 3.4. Figure 4 (open symbols) shows that this placement of the sine edge data provides a good fit with the rest of the data (and this was true also for individual observers—not shown). The value

*n*= 3.4 was obtained by fitting a smooth (Naka-Rushton) curve to each set of data points and then finding the point that minimized the total squared deviation between the sine-wave data and the curves. The fit was good (

*RMS*error = 0.013). Put simply, this means that across the 18 datasets, the sine edge behaved most like a generalized Gaussian edge with an exponent of 3.4, as is evident in Figure 4. At all three scales, Mach bands were much more likely for sine edges (at

*n*= 3.4) than for Gaussian edges (

*n*= 2).

*n*. This correlation was greatest (and the residual squared-error was least) when

*n*= 3.2. The same was true for a comparison of their gradient profiles and also when each waveform was smoothed by a Gaussian whose blur

*b*matched the edge scale

*σ*. Our estimate of

*n*= 3.4, based on similarity of Mach band probabilities, was thus close to the point of maximum physical similarity (

*n*= 3.2).

*n*= 2 to 5) were 4.7, 8.0, and 14.7 min arc from the image center. Mach band position was therefore nearly proportional to the scale of the stimulus waveform.

Filtering scheme | Source | ||

Intrinsic blur | b_{0} | 1.5 min | Band position data; fitted |

E scaling exponent | α | 0.5 | Theory; fixed |

B scaling exponent | β | 1.15 | Band position data; fitted |

E_{2} nonlinearity | p | 3 | Edge position data; fitted |

E_{2} scale | s_{0} | 1.5 min | Fixed = b_{0} |

Filter comparison and noisy decision | |||

E gain | k_{1} | 0.37 | Band probability data; fitted |

B gain | k_{2} | 1 | Fixed |

E_{2} gain | k_{3} | 1.6 | Not used to fit data |

Criterion, scale 3 | c | 1.18 | Band probability data; fitted |

Criterion, scale 6 | c | 1 | Fixed |

Criterion, scale 12 | c | 0.84 | Band probability data; fitted |

Noise, scale 3 | v | 0.18 | Band probability data; fitted |

Noise, scale 6 | v | 0.12 | Band probability data; fitted |

Noise, scale 12 | v | 0.11 | Band probability data; fitted |

*Darth Vader*effect—a shift to the dark side. The origin of this shift in edge positions may well be early compressive nonlinearity in the response to luminance (Mather & Morgan, 1986). Helmholtz noted that this explained why irradiation increases with optical blur—confirmed by Georgeson & Freeman (1997)—and we analyze it quantitatively later. For our data (Experiment 2) the bias was equally evident for edge and bar locations. To allow a better comparison between model and data, this small fixed bias was removed from the data by subtracting from each feature position the mean position of all the features marked for that stimulus scale at

*n*≥ 2. For Figure 5J, 5K, and 5L this simply shifted the spatial origin to the mean marked position in each panel. For Scales 3, 6, and 12 these mean shifts to the dark side were 0.96, 1.25, and 2.12 min arc, respectively.

*n*= 2 to 5, was 5.1, 6.7, and 9.8 min arc at Scales 3, 6, and 12, respectively. Mach bands also became 20%–40% wider as exponent

*n*decreased: At image scales 3, 6, and 12, mean widths were 1.16, 1.39, and 1.44 times wider, respectively, at

*n*= 2 than

*n*= 5.

*x*-axis range is proportional to stimulus scale). Clearly they are similar, but at Scale 3 the bands were (relatively) wider and more separated than at Scales 6 and 12. We show below that a small degree of blur at the input can account for this departure from scale invariance.

*n*for the contrasts used here (0.2, 0.4, 0.8). Mach band probability increased with

*n*,

*F*(6, 24) = 75.2,

*p*< 0.0001, as it did in Experiment 1, but showed only a small dependence on contrast in this suprathreshold range. Mach band probability was similar at contrasts 0.4 and 0.8, but a little weaker at lower contrast (0.2) for

*n*> 2. The main effect of contrast was not significant,

*F*(2, 8) = 1.63,

*p*= 0.255, but the interaction between contrast and exponent (

*n*) was significant,

*F*(12, 48) = 2.9,

*p*= 0.0044. Such near-invariance with respect to contrast is important because the response of any linear filter increases in direct proportion to contrast, and so it follows that Mach band probability cannot depend directly on the response magnitude of linear filters. We consider below a model in which the perception of a bar, or Mach band, depends on the relative activation (the ratio) of even to odd filter responses, which is naturally contrast-invariant.

*n*> 2, Mach bands were less visible at 50 ms than at 100 or 300 ms. This interaction between exponent and duration was highly significant,

*F*(12, 48) = 3.17,

*p*= 0.002, though the main effect of duration was not,

*F*(2, 8) = 3.42,

*p*= 0.085. We show below that the reduction in Mach band probability with brief or lower-contrast images can be explained by an increase in intrinsic blur.

*,*extending a multiscale edge-coding model (N1; Georgeson et al., 2007), but now aiming to capture the location and scale of both bars and edges, using even- and odd-symmetric Gaussian-derivative filters in a scale-space framework.

_{2}(Figure 1D) is introduced to find the edges of Mach bands. All the filter kernels are spatial derivatives of the standard unit-area Gaussian function: where

*s*is the spatial scale.

^{+}) of the N1+ channel to an image

*I*(

*x*) is the convolution of the image with a Gaussian first-derivative filter where

*s*is the scale of the filter, ranging from 1 to 32 min arc in 0.02 octave steps,

*α*= 1/2 is the scale normalization exponent (Lindeberg, 1998), and

*k*is a constant that sets the relative strength of even and odd filter responses. The value of

_{1}*α*is chosen a priori to scale the filter response amplitudes, such that the peak response to a Gaussian edge of blur

*b*occurs in the filter whose scale

*s*=

*b*(Georgeson et al., 2007). This means that the location of the peak response in scale-space encodes edge blur and position veridically, Figure A2(B). An expression for the scale-space edge map of opposite polarity (E

^{−}) is the same except that the ∂ / ∂

*x*filter is of opposite sign.

^{+}of the light-bar channel (N2+) of scale

*s*is defined by convolution with an even-symmetric Gaussian second derivative filter where

*β*is discussed below and

*k*= 1 (fixed). The expression for the scale-space bar map of opposite polarity (B

_{2}^{−}) is the same as B

^{+}except that the filter is of opposite sign. Mathematical analysis, confirmed by numerical computation, showed that choosing

*β*= 3/2 neatly achieves two things: the scale of the bar is veridically encoded (a bar of scale

*b*produces peak response in the B filter that has scale

*s*=

*b*), and the ratio of response amplitudes B/E is scale-invariant. If a signal is spatially magnified by some factor

*m*, then the E and B response patterns are preserved: they are magnified by factor

*m*and shifted in the (log) scale domain by factor

*m*, while the E/B response ratio at (

*m.x, m.s*) after the size change is the same as at (

*x, s*) beforehand. A practical consequence of such scale invariance is that response ratios E/B would be independent of viewing distance. The experiments, however, suggest a slightly lower value for

*β*, discussed below.

^{+}(light bar) and B

^{−}(dark bar) responses, and these responses across many scales

*s*can be visualized as scale-space response maps (Appendix 1). The image of a single light or dark bar produces a peak response at the center of the bar in one of the two maps (B

^{+}, B

^{−}), see Figure A1(A). The coordinates of this peak encode the position and scale of the bar. Its polarity is obtained from the identity of the map (B

^{+}or B

^{−}).

^{+}responses to a light bar are flanked by B

^{−}responses, at all spatial scales, Figure A1(A). Some means is therefore needed for handling these multiple peaks to decide which ones represent visible features, and which do not. The MIRAGE model (Watt & Morgan, 1985), for example, used a set of parsing rules to classify features from filter responses. The N3+ model for edge detection (Georgeson et al., 2007) invokes third order Gaussian derivative filtering in two stages (first derivative followed by second derivative) and half-wave rectification at the output of each stage was designed to screen out spurious peaks and troughs introduced by filtering. For bar detection, however, no such strategy seemed viable, and instead we propose that a comparison between the four response maps may achieve similar ends (Figure A2). Notice how, in Figure A1(C) and A2(C), the peak responses for candidate bars (open squares) flanking a Gaussian edge or Gaussian bar, might be screened out on the grounds that their response amplitudes (B

^{+}, B

^{−}) are lower than the edge responses (E

^{+}, E

^{−}) at the same scale-space point. Those candidate bars are occluded by the edge responses and so might be rejected as visible features. In a similar way, candidate edges (at peaks in the E

^{+}or E

^{−}maps) can be accepted or rejected by examining the edge:bar response ratios at those points. The present experiments support the view that such a comparative selection or decision process operates in human perception of Mach bands (see the section Noisy decision rule below).

*b*

_{0}) was set to 1.5 min arc—small enough to suggest that display blur and dioptric blur (at the eye) were the main sources. We also show (below) that this intrinsic blur increases at lower contrast and short duration (Figure 6). For a further discussion and critique of the concept of intrinsic blur, see Watson and Ahumada (2011).

*β*and the input blur

*b*

_{0}. The assumption of scale invariance of the B:E ratio led us to expect a theoretical value of

*β*= 1.5 (see above) but the data did not support this. The peak filter scales were too large and in consequence the predicted bands were too far apart. A much better fit was obtained with

*β*= 1.15. The choice of blur

*b*

_{0}= 1.5 min arc was largely dictated by Mach band positions at Scale 3 (Figure 5J); smaller values of

*b*

_{0}led the predicted light and dark bands to lie markedly closer together than the observed ones. With these choices for

*β*and

*b*

_{0}, dictated by the data, the model's Mach band positions are shown as blue and yellow lines in Figure 5J, 5K, and 5L. These are close to the mean marked positions (blue and yellow squares). Both the model and observed positions varied little with exponent

*n*, but the model does seem to correctly capture a small increase in Mach band separation at low exponents (

*n*< 3).

^{+}or B

^{−}). If the B:E response ratio is above some criterion

*c*, the observer says “yes”; otherwise “no”. This response ratio is assumed to vary because of added Gaussian noise. Thus the B:E ratio is Gaussian-distributed with mean

*r*and standard deviation

*ν*. The probability of a yes is the area under this distribution that lies above the criterion, and the probability

*p*of a yes is obtained from its

*z*-score where Φ(

*z*) is the standard normal integral. Six parameters were adjusted (see lower half of Table 1; equivalent to two free parameters per stimulus scale) to obtain a good fit by eye between model and observed Mach band probabilities across all three scales (thick grey curves in Figure 5A, 5B, and 5C). Although the parameters were selected to fit the data, it is not trivial that excellent fits were obtained. If the B:E ratio did not rise monotonically with exponent

*n*, no good fit would be possible.

*+*) and odd (Gaussian first derivative, N1+) filters, and then bar features (Mach bands) were found (or rejected) at peaks in the B

^{+}or B

^{−}scale-space response maps by a noisy comparison of B:E responses at those peak points.

*n*= 1, 3, and 5, the yellow and blue curves in Figure 5H show the spatial profiles of responses B

^{+}, B

^{−}at the optimal filter scale; each is a cross-section through the scale-space map. Blue and yellow squares mark the peak B responses, and the probability of a Mach band response depends on the height of that point relative to the edge response (red curve) at the same scale-space point. Moving up to Figure 5E, we see that both the B and E responses at these peak points increase with exponent

*n*but the bar response (B) rises more steeply and so the B/E ratio (

*r*in Equation 5) rises with

*n*. This in turn makes the model's Mach band probability increase with

*n*(thick grey curve, Figure 5B).

*c*and

*v*in Table 1), the model and observed probabilities are in good agreement. Model fits for Scales 3 and 12 are presented in the same way in the left and right columns of Figure 5.

*c*= 0.8 to 1.2) and internal noise (

*v*), the probability of reporting Mach bands can be well described from the way the bar:edge (B/E) response ratio varies with the exponent

*n*. At all three scales the B/E ratio is low (

*r*< 1) when

*n*is low (

*n*< 2) and rises monotonically with increasing

*n*. Mach bands are more probable on Mach ramps (with sharp corners; high

*n*) because the B/E ratio is well above the criterion (

*r*> c), but for Gaussian-like edges (

*n*≈ 2) the bar:edge response ratio is about equal to the decision criterion (giving

*z ≈*0), hence the probability of reporting Mach bands is only about 50%. Despite the apparent similarity in their waveforms, the probability of seeing bands on a sine edge was markedly higher than on a Gaussian edge, and this difference is well captured by the model.

*b*

_{0}. Curves in Figure 6 show the fit of the model, computed exactly as for Figure 5, but allowing intrinsic blur to increase at lower contrasts and briefer durations. In Figure 6A, intrinsic blurs

*b*

_{0}= 3, 1.5, 1 min arc gave excellent fits at contrasts 0.2, 0.4, 0.8, respectively. Similarly, in Panel B, blurs

*b*

_{0}= 3.5, 2.5, 1.5 min arc accounted well for the data at 50, 100, 300 ms.

^{+}and B

^{−}correspond very well with the observed locations of Mach bands (squares in Figure 5J, 5K, and 5L). This depended on an appropriate choice for the scaling exponent

*β*(Equation 4). Nevertheless, despite this success, there remains a substantial problem—the edges of Mach bands. Our observers marked the bands as having edges, with the systematic layout seen in Figure 5J, 5K, and 5L.

*n*. The N3+ model can produce one of the two edges for each Mach band (the inner edge, closer to the center of the image), but only when the corner in the luminance profile is sufficiently sharp, e.g.,

*n*= 10, outside the range of our experiments.

*gradient profile*(rather than the luminance profile) is a Mach ramp, and so the Mach edges are the analog of Mach bands but shifted up by one derivative order; they emerge as peaks in the third derivative rather than the second. But we now see that the edges of Mach bands are a different problem that requires a new solution, as follows.

_{2}

*p*> 1 (here,

*p*= 3) while preserving the sign sharpens and squeezes the B response (black curve in Figure 7C). It then has four (rather than three) points of steepest gradient, and so taking its derivative does lead to four extrema (colored curves, Figure 7C) that correspond well with the observed edge positions (circles).

_{2}is a form of nonlinear third derivative (analogous to, but different from, N3+). As shown in Figure 1D, we take the second derivative map B

^{+}or B

^{−}, pass its (always-positive) values through an accelerating transducer (with power

*p*), filter again with a gradient (Gaussian derivative) operator, half-wave rectify, take the

*p*th root, then find peaks in scale-space as usual using the E

_{2}/B ratio to decide how probable those edges are, via Equation 5. Note that the power

*p*operation is crucial, but the

*p*th root is not so critical, because it does not alter the peak locations introduced by the previous steps. Taking the

*p*th root does, however, render the response linear with respect to contrast, and so makes the E

_{2}/B ratio contrast-invariant.

_{2}response maps are defined by expressions of the form where

*p*= 3,

*k*

_{3}is a constant, and

*s*

_{0}is a small fixed scale that was set equal to the intrinsic blur value (

*s*

_{0}=

*b*

_{0}= 1.5 min arc). The

*E*

_{2}map needs to come in four flavors to capture the positive and negative edges of light bars and dark bars, respectively. These four variants are obtained by using either the B

^{+}or B

^{−}map as input, combined with a positive or negative sign on the ∂ / ∂

*x*filter. Thus a peak in the map of Equation 6 represents the left-hand (light) edge of a light bar. Its right-hand (dark) edge would be captured by peaks in

^{−}rather than B

^{+}as the input map. An interesting consequence of this four-channel scheme (Figure 7D) is that the system not only finds the light and dark edges of Mach bands, but it also knows (from the identity of the channel) what the polarity is and whether those edges arose from, or belong to, a light bar or a dark bar.

^{±}, B

^{±}, and E

_{2}

^{±}{B

^{±}}), along with the features found in those maps, for test images with exponents

*n*= 1, 2, 3, and 5 and scale 6 min arc. The maps for exponents

*n*= 2, 3, and 5 (Figure 8B, 8C, and 8D) show Mach bands (blue and yellow squares), but the bands are weak (open symbols) in the map for

*n*= 1 (Figure 8A). All four maps show the central DL edge (red triangle) that is seen in the data of Figure 5J, 5K, and 5L. The edges of Mach bands (circles) emerge more reliably as

*n*increases.

_{2}mechanism, we can now see in Figure 5J, 5K, and 5L how well the full model (Figure 1D, Figure 8) is able to account for the positions of Mach bands (blue and yellow curves) and their edges (cyan and magenta curves). The goodness-of-fit is clear and the

*RMS*errors are strikingly low, with little or no systematic residual error. The gradual divergence of band and edge positions as

*n*decreases is well described, especially for the dark bands (left side of each panel).

*p*. Raising the power

*p*squeezes the B response profile (Figure 7) even more and so decreases the separation between the edges. For simplicity we set

*p*to a single value (

*p*= 3) and this gave an excellent fit for all three stimulus scales. This proposed nonlinearity is consistent with earlier findings. The value

*p*= 2 represents half-squaring of the linear filter output—a common property of physiological models for V1 cells (e.g., Heeger, 1992)—but higher values of

*p*are common in the physiological and psychophysical literature. Values of

*p*between two and three are typically used to model the response nonlinearity at low contrasts in contrast discrimination experiments (Foley & Legge, 1981; Legge & Foley, 1980; Nachmias & Sansbury, 1974).

*RMS*error between predicted and observed feature locations for Experiment 2, just as in Figure 5, with all parameters unchanged. The goodness-of-fit, assessed by

*RMS*error, is plotted in Figure 9 as a function of the filter scale used. For each stimulus scale (

*σ*) there was a clear point of minimum error, implying an optimum filter scale, and this optimum increased roughly in proportion to the scale of the stimulus.

*α*and

*β*.

*n*= 3, Figure 8C), nearly the same filter scale is picked for all seven features. These filter scales shift to smaller values as exponent

*n*increases (Figure 8D). This is quantified in Figure 9 by the leftward progression of the five grey-filled points that represent the geometric mean filter scale for each of the exponents

*n*= 1 to 5. For

*n*= 1 the filter scales were about an octave higher than the rest, but for

*n*= 2 to 5 (the four leftmost points on each horizontal track), the mean scales clustered quite closely together and the mean of this cluster was, in all three cases, close to the optimum single scale. In short then, in the range

*n*≥ 2 for which the

*RMS*error was computed, the mean filter scale selected by the multiscale model was close to the best single scale. This explains why a fixed scale can substitute for variable scale selection. Importantly, both analyses tell us that to account accurately for the layout of the bar and edge features, the filter scale must increase with the stimulus scale: visual filtering is multiscale.

*L*(

*x*) is the input luminance profile (after blurring by the display and the eye; Figure 1D),

*L*

_{0}is the mean luminance,

*S*is the semisaturation constant, and

*SL*

_{0}is the semisaturation luminance. The constant term (1

*+ S*) is a convenient normalizing factor that implies

*I*= 1 when

*L*=

*L*

_{0}. Note that

*S*is expressed in units of

*L*

_{0}, and so a given value of

*S*represents a certain degree of compressiveness in transduction independently of mean luminance. Lower values of

*S*produce a more compressive response to luminance while higher values tend towards linearity. Georgeson & Freeman (1997) measured and modeled the perceived offset of blurred edges towards their darker side and concluded that a value of

*S*between 0.5 and one was broadly consistent with the observed magnitudes of the offset. Here we chose

*S*= 1.1 to model the data of Experiment 2 and used

*S*= 999 to represent the linear response.

*x*= 0. In contrast, symbols in Figure 10A show experimental data from Experiment 2 now plotted with no correction. Systematic offset of data to the left (the darker side) is obvious, and

*RMS*error in model fitting (1.39 min arc) was more than doubled compared with Figure 5K. But with the inclusion of a nonlinear response to luminance (

*S*= 1.1), the model captured both this offset and the pattern of feature positions very well indeed (Figure 10B).

*RMS*error was low (0.66 min arc) and almost as good as for the hand-crafted correction (0.60 min arc; Figure 5K). This analysis has considered Scale 6, but similar results were obtained at Scales 3 and 12. Averaged over

*N*= 42 features (

*n*≥ 2) the model predicted mean dark shifts of −0.62, −1.06, −2.02 min arc for stimulus Scales 3, 6, 12, and this agreed well with the observed mean shifts of −0.96, −1.25, −2.12 min arc, respectively.

^{−}, solid curves) and the light band (B

^{+}, dashed curves). The compressive luminance response smooths the upper corner of the luminance profile (see Panels E and F), reduces the B

^{+}response and the B

^{+}/E ratio, and so leads to fewer “yes” responses for the light band than the dark band. These predictions (Figure 10D, solid and dashed grey curves) gain some support from the data of Experiment 2, showing that observers marked fewer light Mach bands (light triangles) than dark Mach bands (dark triangles). Such a difference was also evident in the data for Scale 12, but not at Scale 3. A yes-no experiment with more statistical power, many more trials, and independent judgments of light and dark bands would be needed to confirm this finding. Further support comes from earlier findings that increment thresholds (Thomas, 1965) and contrast thresholds (Ross et al., 1989) for seeing the dark bands were systematically lower than for the light bands, implying greater visual sensitivity for the dark bands. Ross et al. (1989) suggested that this difference arose because luminance gain is greater locally for the dark band than for the light band, i.e., that it arose from a form of compressive luminance transduction, as we also suggest.

*σ*= 6 min arc) we can approximate the multiscale model fairly well using a single filter scale (

*s*= 6 min arc) as we saw above. The odd filter is a Gaussian first-derivative, which both smooths the image and computes its derivative (gradient profile). The outcome

*E*(

*x*) is a smoothed derivative, shown as thin lines in Figure 11, for four values of the exponent (

*n*= 1, 1.5, 2, 5). Similarly, the even filter smooths and computes the second derivative of the image (the gradient of the gradient). The outputs

*B*(

*x*) are shown as thick curves in Figure 11, and their peaks—candidate bars—at positions

*x*

_{0}are marked by solid symbols.

*r*=

*B*(

*x*

_{0})/

*E*(

*x*

_{0}) was noisy, with variance

*v*

^{2}at scale

*σ*, and that Mach bands would be seen on those occasions when the ratio

*r*exceeded some criterion

*c*. To visualize this noise in Figure 11, we can pretend that the odd filter is noise-free and attribute all the noise to the even filter. Thus the grey bars show ±1 standard deviation of the noise, equal to

*v*.

*E*(

*x*) when expressed this way. The clear message is that for

_{0}*n*= 1 the candidate bar response lies well below the edge response and predicts 0% Mach bands, but as

*n*increases the peak bar response rises and eventually exceeds the edge response quite reliably, giving 94% Mach bands for

*n*= 5. This trend is similar to the experimental results (Figures 5B and 6). The mean Mach band position (±8 min arc) is also well described, as is the increase in separation of the bands as

*n*decreases.

*α*,

*β*,

*k*

_{1}, Table 1) and a suitably selected spatial scale

*s*for the filters. We see from Figure 11 that a single-scale model contains all the main elements for our proposed account of Mach bands, but automatic selection of spatial scale is a key advantage of the multiscale model, to which we now turn.

*n*that controls the sharpness of the corner in the luminance profile. This main result (Figures 4 and 6) is similar to Ross et al.'s (1989) finding that contrast sensitivity for seeing Mach bands decreased markedly as the ramp was made more blurred. Blurring the ramp is analogous to decreasing

*n,*and in both cases the second-derivative amplitude is correlated with corner sharpness and with Mach band perception. But, in apparent conflict with this idea, Experiments 1 and 3 showed that the likelihood of seeing Mach bands was largely independent of the spatial scale and image contrast in the suprathreshold range. These near-invariances imply that the probability of seeing Mach bands is not determined directly by the amplitude of peaks in the second derivative, because that is not invariant—it increases with image contrast and decreases greatly with increasing image scale (Figure 3).

*n*. Second, the profound decrease in second derivative amplitude that would occur with increasing stimulus scale (Figure 3) is countered by the scale normalization factor (

*s*, Equation 4). This key factor increasingly amplifies the response of the larger-scale filters. Importantly, the B:E ratio—and hence Mach bands—would be precisely scale-invariant as well as contrast-invariant, if

^{β}*s*=

^{β}*s.s*, implying

^{α}*β*=

*α*+ 1. This is because the second-derivative falls much faster than the first-derivative with increasing scale (inverse-square, vs. inverse), and so to maintain a constant B:E ratio, it needs to be amplified correspondingly more (by factor

*s*). Ideally then,

*α*= 0.5 and

*β*= 1.5 (see Appendix 2). We set

*α*= 0.5, but found that

*β*= 1.15 (rather than 1.5) gave a better account of the data on Mach band layout, and this necessarily compromised the B:E scale invariance to a small extent that was compensated by allowing the criterion response ratio to differ at different stimulus scales. It would be more elegant if perfect scale invariance prevailed (

*β*= 1.5), but the data dictated otherwise.

_{2}) that is a nonlinear form of third derivative alongside the basic edge response (E) that is based on the first derivative. Such duplication is not implausible, but on grounds of parsimony, having two edge mechanisms might sound like one too many. We wonder whether some additional nonlinearity in the N3+ channel (Georgeson et al., 2007) might be able to unify the properties of E and E

_{2}into a single edge mechanism, but this remains a task for future work.

*multimodel inference*have been developed (Burnham & Anderson, 2004). Full evaluation of other models is beyond the scope of this paper, but some brief comments are in order.

^{+}and B

^{−}in the present model; then it combines outputs of the same sign across scales and uses a fixed set of rules to interpret the two resulting profiles as features. In brief, if two adjacent peaks of the combined responses have opposite signs they are interpreted as an edge, but if those two peaks are separated by a gap (a null region in the filter responses) then the two peaks would be interpreted as a pair of light and dark bars. The response to a Mach ramp is of the latter kind, giving two bars as the output description. This rule works well for Mach ramps (high

*n*), but it cannot predict Mach bands on a Gaussian edge, because no null region exists between the peak and trough in this case at any one filter scale or in the combined response. Our experiments consistently showed Mach bands to be reported on about 60% of trials for Gaussian edges (

*n*= 2), and this does not appear to be a baseline rate of false alarms or guesses because much lower probabilities were recorded for

*n*= 1.5 and

*n*= 1. Thus, without modification, MIRAGE cannot predict the Mach bands seen on Gaussian edges. Indeed the two interpretations (as one edge or two bars) are mutually exclusive; unlike the present model, MIRAGE could never see Mach bands and an edge at the same time. This is evidence against the enforced combination of responses across filter scale, as Kingdom and Moulden (1992, p. 1579) pointed out.

*σ*for high

*n*, was always less than 25 min arc). However, our own simulations of MIRAGE confirmed the original report of Watt and Morgan (1985), reiterated by Morgan and Watt (1997), that MIRAGE does deliver Mach bands for ramps as narrow as 5 min arc. We found that this depended on a suitable choice of the model's noise-reduction threshold. Thus the apparently conflicting predictions derived from the same model may hinge on details of implementation—an example of the difficulty of model comparison referred to above. When we applied MIRAGE to our stimulus set we confirmed that no Mach bands were predicted for

*n*≤ 2, and we could not find any set of parameters that would consistently deliver Mach bands at all three stimulus scales (3, 6, and 12 min arc).

*n*≤ 2, no Mach bands appeared; instead, at all filter scales, a single energy peak represented only the central edge. For

*n*= 3 to 5, Mach bands of the correct polarity appeared across a range of filter scales, but their positions varied smoothly with filter scale and without a rule for scale selection we could not properly compare those positions with our data. For the sine edge, Mach bands were predicted only over a narrow range of filter scales. At finer scales the energy model instead incorrectly reported edges (not bars) located at

*x*= ±

*h*/ 2, where

*h*is the width (half-period) of the sine edge—an effect that arises from discontinuity in the higher spatial derivatives at these two locations. To screen out these false edges a rule for scale selection would again be needed. In short, the energy model does predict Mach bands, but would need modification to be tested against our results. As with MIRAGE, it is possible that some new assumptions about scale selection, filter bandwidth, or other changes of implementation detail might improve it.

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- Obtain the data for one trial.
- If there are seven features, place each one in the seven bins, in position order.
- If there are fewer than seven features, then:
- a. Place any bars into the bar bin of its polarity (Bins 2 and 6).
- b. Place any DL edges that are to the left of the image center in Bin 1.
- c. Place any DL edges that are to the right of the image center in Bin 7.
- d. If there are three LD edges, place them in Bins 3 to 5, in position order, but if there are less than three then delay assignment until second pass.
- Repeat the above steps for every trial in the same condition.

_{2}process is added, we refer to “the full N2+1+ model.”

^{+}and B

^{−}) is shown in Figure A1(A) in response to two Gaussian bars of different scales. The corresponding edge maps alone (E

^{+}and E

^{−}) are in Figure A1(B). Since all responses are positive, a convenient way to visualize the population response is to plot a single map,

*M*(

*x, s*) = max(E

^{+}, E

^{−}, B

^{+}, B

^{−}) where the max operator picks the largest of the four responses at each position and scale (

*x, s*). The composite map

*M*(

*x, s*) for the same two bars (Figure A1[C]) shows how the relation between E and B responses may be important in deciding what features are present. Potential Mach bands (blue squares in Figure A1[A]) may be occluded by larger edge responses (E

^{+}or E

^{−}) at the same place and might not be visible (squares now shown as open symbols in Figure A1[C]), depending on noise and decision factors discussed in the main text.

*α*= 0.5,

*β*= 1.5, the relative activation of E and B filters and the relative scales and positions of bar and edge features are all scale-invariant. Despite these elegant properties, we found that that this scale invariance property did not hold exactly when the experimental data were fitted; we had to reduce

*β*to about 1.15 to match the observed spatial layout of Mach bands shown in Figure 5, thus compromising this scale invariance to some extent.